Koch's Siamese

It is a fractal, an elongated variant of the most famous Koch snowflake.

File:Siamese snowflakes (Tartapelago).gif

The Koch's siamese

Siamese and anti-siamese

File:Snoflake&antisnowflake.jpg

Comparison between Koch's snowflakes and anti-snowflake

File:KochSiamese&Kochantisiamese.jpg

Comparison between Koch's Siamese and anti-Siamese

The Siamese is obtained from the Koch curve by lacing externally a rhombus with angles 60, 120, 60, 120 (degrees), i.e. such that its smaller diagonal divides it into two equilateral triangles. If instead you lace the four sides of the rhombus internally you obtain the antisiamese. The perimeter of the Siamese, like that of the bow, is infinite being composed of Koch curves. The area of the Siamese increases by 40% compared to that of the turbot. That of the anti-Siamese decreases by the same percentage. The Siamese was introduced by Giorgio Pietrocola in 2020 with the publication on Maecla of the Art of plane tessellation with Koch flakes

Rep-tile

File:Frattale infinito rep-tile.gif

$Rep-\mathrm{\infty }$

If we lace the rhombus already described both internally and externally, the property of the Siamese of being able to decompose into infinite miniature copies of itself is highlighted. It is therefore a replicant of infinite order.

Koch Siamese Tessellations

It can be obtained in many ways, but the sizes of the siamese are infinite and decrease in geometric progression.

• 
  MSWLogo program
• 
  tessellations with siamese
• Error creating thumbnail:
  with anti-siamese
• Error creating thumbnail:
  with anti-snowflakes
• Error creating thumbnail:
  variation on the theme

References

• Helge von Koch (1904). Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire (in français). Search this book on
• Pietrocola, Giorgio (2023). "Il siamese di Koch. Un frattale straordinariamente vario nel tassellare il piano" (PDF). Periodico di Matematica. Accademia di Filosofia delle Scienze Umane. (IV) Vol. V(2): 109–124. ISSN 2612-6745.
• Giorgio Pietrocola (2023). Riflettere sulla didattica della Matematica per insegnare: Ricerche ed esperienze. Bonomo. ISBN 978-88-6972-305-6. Search this book on
• Pietrocola, Giorgio (2023). "Dal merletto di Koch alla tassellazione frattale del piano con una sola figura frattale". Le Monnier: 194–203. ISSN 0390-5543.

External links

• Pietrocola, Giorgio (2020). "Arte della tassellazione del piano con fiocchi di Koch in Tartapelago". Maecla.

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